Webs of stars or how to triangulate free sums of point configurations

TL;DR

This paper introduces new combinatorial tools called 'web of stars' and 'stabbing poset' to classify triangulations of free sums of point configurations, with applications to smooth Fano polytopes.

Contribution

It develops a novel classification framework for triangulations of free sums using two new partially ordered sets, advancing the understanding of polytope triangulations.

Findings

Triangulations of free sums are classified via summand triangulations.

New posets help analyze triangulations with marked points.

Application to smooth Fano polytopes demonstrates the method's utility.

Abstract

The triangulations of point configurations which decompose as a free sum are classified in terms of the triangulations of the summands. The methods employ two new kinds of partially ordered sets to be associated with any triangulation of a point set with one point marked, the "web of stars" and the "stabbing poset". Triangulations of smooth Fano polytopes are discussed as a case study.